Generalized monotonicity from global minimization in fourth-order ODEs

We consider solutions of the stationary Extended Fisher-Kolmogorov equation with general potential that are global minimizers of an associated variational problem. We present results that relate the global minimization property to a generalized concept of monotonicity of the solutions. This monotonicity can be described as the lack of intersections of the solution curve when projected onto the $(u,u')$--plane. Our method is based on applying a cut-and-paste argument in the space $H^2( {mathbb{R )$ to intersections in the $(u,u')$--plane. The statements and proofs are presented for the Extended Fisher-Kolmogorov equation, but the method can be directly extended to a wide class of fourth-order ordinary differential equations that derive from minimization problems.